Question

prove that : ( 1 - sin2 theta ) sec2 theta =1

Answer 1

(1- sin^2 theta )sec ^2 theta
=( cos ^ 2 theta ) 1/ cos ^2 theta [[sin ^theta + cos^2 theta =1. and. sec ^2 theta = 1/ cos^2 theta]]
=1

Answer 2

Proof for : ( 1 - sin2 theta ) sec2 theta =1 is given as follows

To prove,

$(1-sin^2\theta)sec^2\theta = 1$

we know the trigonometric properties,

$sin^2\theta + cos^2\theta = 1$

$\Rightarrow 1-sin^2\theta = cos^2\theta$

Therefore, we have,

LHS:

$= (1-sin^2\theta)sec^2\theta$

$= (cos^2\theta)sec^2\theta$

we know the trigonometric ratios, given by,

$cos\theta = \dfrac{1}{sec\theta}$

$\Rightarrow sec\theta=\dfrac{1}{cos\theta}$

$= (\dfrac{1}{sec^2\theta})sec^2\theta$

= 1

:RHS

Hence the proof.